566 Transactions of the Royal Society of South Africa. 
the distance when the spring hangs in equilibrium under its own weight 
and that of M. 
T is the tension of the spring in the neighbourhood of the element. 
The element originally of length Ax is stretched to a£ under tension T. 
Ai-Ax_T T 
• • Ax -E dx-^'^^ 
The equation of motion of the element is 
/ ^T 
fj . Ax . 'i=pg . Ax-h{T + ~ . Ax^-T 
i.e. ,i = ,g + 'j^=„g + ^'^. 
Since to marks an equilibrium position 
.*. if z be the displacement of the element from its equilibrium position 
z=l—l^, and we have 
^x- 
Try a solution of the type ;s=X pt where X is a function of x only. 
.-. -pi>'-^X=E-^ .-. X=A cos xp^^ 1^ +B sin xp^ ^ 
The condition 2=0 when x—O for all time cuts out the cosine terms 
in x. 
If the system is started from rest (as happens in the ordinary experi- 
ment), 2=0 when ^=0 for all values of x; this cuts out the cosine 
term in t. 
We are left with 
A sin pt sin px ^ 
The value of p is now got by considering the conditions at the free end. 
The element Ax is stretched to length A£o i^ii<3.er the tension 
pg{l-x)-\-Mg. 
Al, — Ax _ pg{l—x)-Y'M.g 
Ax ~ E 
gplx 1 gpx^ 'M.gx 
Then 
T=E(f-^-l)=E(i^+^»-l) 
\^x ! ^x I 
, the end value of T is 
